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Register a new userWeil sums of binomials: properties, applications, and open problems
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Added by
Daniel Katz
Publication date
2018
fields
Informatics Engineering
and research's language is
English
Authors
Daniel J. Katz
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We present a survey on Weil sums in which an additive character of a finite field $F$ is applied to a binomial whose individual terms (monomials) become permutations of $F$ when regarded as functions. Then we indicate how these Weil sums are used in applications, especially how they characterize the nonlinearity of power permutations and the correlation of linear recursive sequences over finite fields. In these applications, one is interested in the spectrum of Weil sum values that are obtained as the coefficients in the binomial are varied. We review the basic properties of such spectra, and then give a survey of current topics of research: Archimedean and non-Archimedean bounds on the sums, the number of values in the spectrum, and the presence or absence of zero in the spectrum. We indicate some important open problems and discuss progress that has been made on them.
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We investigate the $p$-adic valuation of Weil sums of the form $W_{F,d}(a)=sum_{x in F} psi(x^d -a x)$, where $F$ is a finite field of characteristic $p$, $psi$ is the canonical additive character of $F$, the exponent $d$ is relatively prime to $|F^times|$, and $a$ is an element of $F$. Such sums often arise in arithmetical calculations and also have applications in information theory. For each $F$ and $d$ one would like to know $V_{F,d}$, the minimum $p$-adic valuation of $W_{F,d}(a)$ as $a$ runs through the elements of $F$. We exclude exponents $d$ that are congruent to a power of $p$ modulo $|F^times|$ (degenerate $d$), which yield trivial Weil sums. We prove that $V_{F,d} leq (2/3)[Fcolon{mathbb F}_p]$ for any $F$ and any nondegenerate $d$, and prove that this bound is actually reached in infinitely many fields $F$. We also prove some stronger bounds that apply when $[Fcolon{mathbb F}_p]$ is a power of $2$ or when $d$ is not congruent to $1$ modulo $p-1$, and show that each of these bounds is reached for infinitely many $F$.
We investigate the Renyi entropy of independent sums of integer valued random variables through Fourier theoretic means, and give sharp comparisons between the variance and the Renyi entropy, for Poisson-Bernoulli variables. As applications we prove that a discrete ``min-entropy power is super additive on independent variables up to a universal constant, and give new bounds on an entropic generalization of the Littlewood-Offord problem that are sharp in the ``Poisson regime.
We revisit a recent bound of I. Shparlinski and T. P. Zhang on bilinear forms with Kloosterman sums, and prove an extension for correlation sums of Kloosterman sums against Fourier coefficients of modular forms. We use these bounds to improve on earlier results on sums of Kloosterman sums along the primes and on the error term of the fourth moment of Dirichlet $L$-functions.
Francis Castro, et al [2] computed the exact divisibility of families of exponential sums associated to binomials $F(X) = aX^{d_1} + bX^{d_2}$ over $mathbb{F}_p$, and a conjecture is presented for related work. Here we study this question.
Arithmetic dynamics is the study of number theoretic properties of dynamical systems. A relatively new field, it draws inspiration partly from dynamical analogues of theorems and conjectures in classical arithmetic geometry, and partly from $p$-adic analogues of theorems and conjectures in classical complex dynamics. In this article we survey some of the motivating problems and some of the recent progress in the field of arithmetic dynamics.
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